cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A294655 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-2))*(1 - x^(2*k))^(k*(3*k+2))).

Original entry on oeis.org

1, 1, 6, 14, 45, 106, 290, 683, 1698, 3918, 9179, 20640, 46444, 101819, 222092, 475886, 1012270, 2124725, 4425195, 9118705, 18648048, 37797126, 76062443, 151889787, 301296200, 593593192, 1162276735, 2261819285, 4376578818, 8421295585, 16118902083, 30694325652, 58164428059
Offset: 0

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Author

Ilya Gutkovskiy, Nov 06 2017

Keywords

Comments

Euler transform of the generalized octagonal numbers (A001082).

Links

Crossrefs

Cf. A000294, A001082, A274998, A294591, A294622, A294654, A294691, A294692.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 2)) (1 - x^(2 k))^(k (3 k + 2))), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d^2 + d - Ceiling[d/2]^2), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001082(k+1).
a(n) ~ exp(Pi * 2^(3/2) * n^(3/4) / (3*5^(1/4)) + 3*Zeta(3) * sqrt(5*n) / (2*Pi^2) - (45*Zeta(3)^2 / Pi^5 + Pi/6) * 5^(1/4) * (n^(1/4) / 2^(5/2)) + 225 * Zeta(3)^3 / (4*Pi^8) - Zeta(3) / (32*Pi^2) + 1/8) * Pi^(1/8) / (A^(3/2) * 2^(77/48) * 5^(5/32) * n^(21/32)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 07 2017

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal numbers.

Links

Crossrefs

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).

Original entry on oeis.org

1, 5, 31, 148, 667, 2754, 10823, 40393, 145085, 502780, 1690603, 5530649, 17658430, 55141520, 168751779, 506933980, 1496999360, 4350994324, 12460305177, 35192973824, 98116587875, 270220568883, 735668636567, 1981082952258, 5279879097853, 13933764841202
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2017

Keywords

Links

Crossrefs

Cf. A074962, A274998, A278768, A294655, A294667.

Programs

  • Maple
    N:= 50:
S:= series(mul(1/(1-x^k)^(k*(3*k+2)), k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 07 2017
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) + 2*Zeta(3) * sqrt(5*n) / Pi^2 - 2*5^(5/4) * Zeta(3)^2 * n^(1/4) / Pi^5 + 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) + 1/6) * Pi^(1/6) / (A^2 * 2^(3/2) * 5^(1/6) * n^(2/3)), where A is the Glaisher-Kinkelin constant A074962.

A317021 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k-1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 11, 51, 216, 861, 3477, 13367, 50377, 184667, 664484, 2345230, 8142476, 27825576, 93750686, 311682789, 1023547782, 3322634928, 10669887669, 33916213669, 106776876109, 333111724130, 1030264525744, 3160359629535, 9618807643826, 29057370625281, 87153154537437
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 19 2018

Keywords

Comments

Euler transform of A002419.

Links

Crossrefs

Cf. A000391, A002419, A274998, A279219, A305653, A317017, A317019, A317020.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add(
      (3*d-1)*binomial(d+2, 3)/2*d, d=numtheory
      [divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 19 2018
  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^((3 k - 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 26; CoefficientList[Series[Exp[Sum[x^k (1 + 5 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d - 1)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002419(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 5*x^k)/(k*(1 - x^k)^5)).
a(n) ~ 1/(2^(1987/2160) * 3^(713/1080) * 7^(173/2160) * n^(1253/2160) * Pi^(7/360)) * exp(-1/72 + (1/12-Zeta'(-1))/6 - Zeta(3)/(30 * Pi^2) + (111 * Zeta(5))/(200 * Pi^4) - (7056 * Zeta(3) * Zeta(5)^2)/Pi^12 - (592704 * Zeta(5)^3)/(5 * Pi^14) + (43016085504 * Zeta(5)^5)/(5 * Pi^24) + (2 * Zeta'(-3))/3 + ((-7 * (7/2)^(1/6) * Pi)/(3200 * 3^(2/3)) + (14 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/Pi^7 + (1029 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(5 * Pi^9) - (17978688 * 2^(5/6) * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/Pi^19) * n^(1/6) + (-((7/6)^(1/3) * Zeta(3))/(2 * Pi^2) - (7 * 3^(2/3) * (7/2)^(1/3) * Zeta(5))/(5 * Pi^4) + (75264 * 6^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/2) * Pi)/60 - (1008 * sqrt(14) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((6 * 6^(1/3) * 7^(2/3) * Zeta(5))/Pi^4) * n^(2/3) + ((2 * (2/7)^(1/6) * 3^(2/3) * Pi)/5) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018
Showing 1-4 of 4 results.

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